Beyond the standard gauging: gauge symmetries of Dirac sigma models

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Published for SISSA by Springer

Received: July 8, 2016 Accepted: August 10, 2016 Published: August 31, 2016

Beyond the standard gauging: gauge symmetries of Dirac sigma models

Athanasios Chatzistavrakidis,^a Andreas Deser,^b Larisa Jonke^c and Thomas Strobl^d

aVan Swinderen Institute for Particle Physics and Gravity, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands

bInstitut f¨ur Theoretische Physik, Leibniz Universit¨at Hannover, Appelstr. 2, 30167 Hannover, Germany

cDivision of Theoretical Physics, Rudjer Boˇskovi´c Institute, Bijeniˇcka 54, 10000 Zagreb, Croatia

dInstitut Camille Jordan, Universit´e Claude Bernard Lyon 1,

43 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France E-mail: a.chatzistavrakidis@rug.nl,

andreas.deser@itp.uni-hannover.de,larisa@irb.hr, strobl@math.univ-lyon1.fr

Abstract:In this paper we study the general conditions that have to be met for a gauged extension of a two-dimensional bosonic σ-model to exist. In an inversion of the usual ap- proach of identifying a global symmetry and then promoting it to a local one, we focus directly on the gauge symmetries of the theory. This allows for action functionals which are gauge invariant for rather general background fields in the sense that their invariance conditions are milder than the usual case. In particular, the vector fields that control the gauging need not be Killing. The relaxation of isometry for the background fields is con- trolled by two connections on a Lie algebroidL in which the gauge fields take values, in a generalization of the common Lie-algebraic picture. Here we show that these connections can always be determined whenLis a Dirac structure in theH-twisted Courant algebroid.

This also leads us to a derivation of the general form for the gauge symmetries of a wide class of two-dimensional topological field theories called Dirac σ-models, which interpo- late between the G/G Wess-Zumino-Witten model and the (Wess-Zumino-term twisted) Poisson sigma model.

Keywords: Gauge Symmetry, Sigma Models, Differential and Algebraic Geometry ArXiv ePrint: 1607.00342

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Contents

1 Introduction 1

2 General gauge invariant 2D σ-models 5

2.1 Gauge theory and invariance conditions for minimally-coupled gauge fields 5 2.2 Geometric interpretation and generalized Riemannian metric 7 2.3 Inclusion of WZ term and non-minimally coupled gauge fields 9 3 Connections and gauge symmetries for Dirac structures 11

3.1 Dirac structures and Diracσ-models 11

3.2 Constructing the connections ∇^± on Dirac structures 13

3.3 General form of the gauge symmetries for DSMs 15

4 Application: Wess-Zumino Poisson σ-model 16

5 Summary of results and conclusions 19

A A direct proof for the gauge symmetries of the PSM 21

B Structure functions for Dirac structures 23

1 Introduction

When does a problem, for example a variational one with some associated action functional, have a local symmetry? This is an interesting question due to the importance of local sym- metries in nearly every corner of theoretical physics. Gauge theories exhibiting some local symmetry are usually related to the introduction of spacetime 1-forms (gauge fields) taking values in some Lie algebrag. This procedure has as starting point the identification of some global symmetry of the theory which is then promoted to a local one with the aid of these gauge fields. However, from a mathematical perspective, symmetries are associated to more general structures than groups and algebras. These structures are groupoids and algebroids (see the inspiring ref. [1]). The purpose of the present paper is to study some aspects of the role of Lie algebroid structures in the simple physical setting of two-dimensional bosonic σ-models. To this end we first present some material from our upcoming, more mathemati- cally oriented work [2], as a basis and motivation for the new results presented in this paper.

Two-dimensional bosonic σ-models are based on maps X = (Xⁱ) : Σ → M, i = 1, . . . ,dimMfrom a two-dimensional source space Σ of Lorentzian or Euclidean signature to a target spaceM. These theories are interesting because they describe strings propagating

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in target spacetimesM through the scalar fieldsXⁱ.¹ The source space is equipped with a metric h= (hαβ), α, β= 0,1, which is a dynamical field in the theory. Furthermore, since the massless NSNS spectrum of string theory contains a Riemannian metric g = (gij) on the target spaceM, an antisymmetric Kalb-Ramond 2-formB = (Bij) and a scalar dilaton Φ, the relevant bosonic σ-model is

S[X, h] = 1 4πα^′

Z

Σ

1

2gij(X)dXⁱ∧∗dX^j+ 1 4πα^′

Z

Σ

1

2Bij(X)dXⁱ∧dX^j+ 1 8π

Z

Σ

RΦ∗1, (1.1) where R is the world sheet curvature scalar.² This action depends on the world sheet metric h through the Hodge star operator ∗, with∗² =∓1 for Euclidean and Lorentzian signature, respectively. Our first purpose then is to find a gauged extension of this action functional for as general background fields g(X) and B(X) as possible.

The motivation to consider gauged versions of theσ-model is twofold. The first moti- vation is the intention to derive new theories on quotient spaces [2], similarly to the case of gauging the Wess-Zumino-Witten (WZW) model whose target space is a group G [3].

When the gauging is along (the adjoint action of) a subgroupHofGone obtains theG/H WZW models,G/G being an extremal case [4,5]. Another motivation to consider gauged σ-models is the study of target space duality in string theory. Recall that the celebrated Buscher rules relating two dual string backgrounds with Abelian isometries are derived by considering an intermediate gauge theory which reduces to the two dual actions upon inte- grating out different sets of fields, be they additional scalar fields (Lagrange multipliers) or the (non-dynamical) gauge fields that are coupled to the theory [6,7]. A similar procedure may be followed in the case of non-Abelian isometries [8–11].

An important difference of our procedure with respect to the traditional gauging is that we follow an inverted logic. Instead of identifying global symmetries and then promoting them to local ones, we are looking for more general gauge invariant extensions of the action functional (1.1) for arbitrary background fields. The crucial difference is then that the invariance conditions for the background fields imposed in the case of a global symmetry are relaxed and replaced by much milder ones. This point of view was originally introduced in [12] and studied further in [13–16].

The above procedure amounts to a threefold generalization of the standard picture of gauged σ-models with g-valued gauge fields, minimally-coupled to the scalar degrees of freedom Xⁱ and with strong invariance conditions on the background fields. The first generalization is that we allow the gauge fields to take values in some Lie algebroid L instead of a Lie algebra.³

A clear and physically motivated way to introduce this structure goes through the well-known Cartan problem. Following the lecture notes [19], in this problem we consider two functionsC_bc^a andeⁱ_a defined on an open setU ⊂Rⁿ, witha= 1, . . . , randi= 1, . . . n.

1For the purposes of the present paper we are not going to consider fermions and supersymmetry but rather focus on the classical bosonic theory.

2Since the dilaton coupling contributes at one-loop rather than at leading order it will be ignored in most parts of the analysis, which will remain classical in this paper.

3This idea goes back to [17,18] and, as seen only recently [2], can be generalized to just anchored bundles covering a foliation.

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Then the aim of the problem is to determine (i) a manifold N, (ii) a set of 1-forms {e^a} being a coframe on N, and (iii) a map ϕ : N → U ⊂ Rⁿ, such that the following two equations hold:

de^a = −1

2C_bc^a(ϕ)e^b∧e^c, (1.2)

dϕⁱ = eⁱ_a(ϕ)e^a . (1.3)

Taking the exterior derivative on each side of each equation we directly obtain two necessary conditions,

C_e[b^a C_cd]^e =eⁱ_[b∂iC_cd]^a , (1.4) 2e^j_[b∂_jeⁱ_c] =C_bc^aeⁱ_a, (1.5) where here and in the following symmetrizations and antisymmetrizations of indices are taken with the corresponding weight. In the special case that the coefficients C_bc^a are constant, the right hand side of the first condition vanishes and the condition simply becomes the usual Jacobi identity for a Lie algebrag. Moreover, we then can define theg- valued Maurer-Cartan forme=e^aea∈Ω¹(N,g) which satisfies the famous Maurer-Cartan equation

de+1

2[e, e] = 0. (1.6)

However, when C_bc^a are not constant, but ratherϕ-dependent functions, one may address the problem in the following way, cf. [19]. First one considers a vector bundle L of rankr overU. Lis equipped with a Lie bracket [·,·]Land C_bc^a become structure functions for this bracket, namely

[ea, eb]L=C_ab^c (ϕ)ec, (1.7) for a basis of local sections ea ∈ Γ(L). Furthermore, a smooth map ρ : L → T U is introduced, mapping sections of Lto vector fields. The bracket on L satisfies the Leibniz identity [e, f e^′]L= f[e, e^′]L+ρ(e)f e^′, for e, e^′ ∈Γ(L) and f ∈C^∞(U). Then eq. (1.5) is simply the statement that this map is a homomorphism:

(1.5) ⇔ ρ([ea, e_b]L) = [ρ(ea), ρ(e_b)]. (1.8) Moreover, eq. (1.4) simply states that:

(1.4) ⇔ Jac(a, b, c) := [ea,[e_b, ec]_L]_L + (cyclic permutations) = 0 ; (1.9) in other words it is the Jacobi identity for the Lie bracket onL. The above triple (L,[·,·]_L, ρ) is an example of a Lie algebroid. More generally, a Lie algebroid over a manifoldM consists of a vector bundleLoverM, equipped with a Leibniz-Lie bracket on its sections Γ(L) and a bundle map ρ : L → T M. Lie algebras are included in the definition; they are the corresponding structures in the case that the base manifold is just a point.

Returning to our discussion of theσ-model (1.1), let us take a step back and recall that given a set of vector fieldsρa=ρⁱ_a(X)∂i, the action (1.1) has a global symmetry of the form

δǫXⁱ =ρⁱ_a(X)ǫ^a (1.10)

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for rigid transformation parametersǫ^aonly under the following three invariance conditions:

Lρag= 0, LρaB = dˆθa, LρaΦ = 0, (1.11) for some 1-form ˆθa. Then ρa are Killing vector fields generating isometries for the metric g. They can be non-Abelian and satisfy an algebra

[ρa, ρb] =C_ab^c ρc .

The symmetry is gauged throughg-valued 1-formsA= (A^a). The gauge parametersǫ^anow depend on the world sheet coordinates (σ^α) and the gauge fields transform according to

δǫA^a= dǫ^a+C_bc^aA^bǫ^c, (1.12) which is the standard infinitesimal gauge transformation for non-Abelian gauge fields.

Clearly, the above invariance conditions should also hold at the level of the gauged model, and in addition ˆθa has to be zero. The latter fact is related to the validity of minimal coupling and will be discussed below.

However, (1.11) are too strong and restrictive. We address this issue by following a more unorthodox route and asking the question

• Under which invariance conditions does the action functional (1.1) have a local sym- metry δǫXⁱ=ρⁱ_a(X)ǫ^a(σ) at leading order inα^′?

As already mentioned, the main difference is that we do not require a global symmetry to start with, thus allowing for more general background fields. To answer this question we are going to consider gauge fields taking values in a Lie algebroid

L→^ρ T M . (1.13)

The analysis reveals that the invariance conditions (1.11) are replaced by the milder ones [2]

L_ρag = ω_a^b∨ι_ρbg−φ^b_a∨ι_ρbB , (1.14) LρaB = ω_a^b∧ιρ_bB±φ^b_a∧ιρ_bg , (1.15) where∨is defined as the symmetric productα∨β =α⊗β+β⊗αand∧as the antisymmetric oneα∧β =α⊗β−β⊗α, whileω_a^b =ω_ai^b dxⁱ andφ^b_a =φ^b_aidxⁱare 1-forms onM, where (xⁱ) is a local coordinate system on it. In a geometric interpretation of these relations we find thatω_a^b and φ^b_ain fact define two connections∇^± onL. This is our second generalization.

Next let us return to the issue of minimal coupling. As mentioned above, when ˆθa

in (1.11) is not zero, minimal coupling is not enough. This is certainly not a new obser- vation. It is related to the presence of Wess-Zumino (WZ) terms and has been clarified in many works starting from [8,9,20] and more recently in [21,22]. One of the main points is that when there is a WZ term one has to add terms in the gauged action which contain the gauge fields outside the covariant derivatives that are formed through minimal coupling.

Then the gauging is obstructed by additional constraints on top of the previously men- tioned invariance conditions. Revisiting this issue in the light of the two generalizations

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we already mentioned leads us again to a set of milder conditions and constraints for WZ terms. Furthermore, it turns out that the target space of the gauged theory is lifted to the generalized tangent bundle T M⊕T^∗M [2].⁴

A very interesting question arising from the above results is whether and when the new and milder invariance conditions for the gauging of the σ-model can be solved. We will show that whenLis a Dirac structure, i.e. a particular type of Lie algebroid associated to the standard Courant algebroid onT M⊕T^∗M twisted by a closed 3-formH, then the two Dirac structure connections solving the invariance conditions can be determined. Indeed we are going to provide general closed expressions for them. This is the central result of this paper. Additionally, it will allow us to find the form of the gauge symmetries for Dirac σ-models, which are very general two-dimensional topological field theories interpolating between WZW models and Poisson σ-models and they were introduced in [23, 24]. Our analysis will be complemented with an explicit example where we will apply our findings to the WZ Poisson σ-model with a kinetic term.

2 General gauge invariant 2D σ-models

In this section we present some of the material from [2] in a self-contained manner, since it is needed as a basis for the subsequent sections. This applies in particular to sections 2.1 and 2.3. Section 2.2, on the other hand, provides some relation of the obtained formulas to generalized geometry; for the complete picture of this we refer to [16].

2.1 Gauge theory and invariance conditions for minimally-coupled gauge fields As explained in the introduction, our starting point is the bosonic sector of closed string theory at leading order in the string slope parameterα^′. This is described by the following σ-model action:

S0[X] = Z

Σ

1

2gij(X)dXⁱ∧ ∗dX^j + Z

Σ

1

2Bij(X)dXⁱ∧dX^j, (2.1) where Σ is the 2D world sheet,X= (Xⁱ) : Σ→M, i= 1,· · · , n= dimM,is the map from the world sheet to the target space M and ∗ denotes the Hodge duality operator on the world sheet. Here and in the following we work in units where 4πα^′ = 1. The background fields g = g_ijdxⁱ⊗dx^j = ¹₂g_ijdxⁱ∨dx^j and B = ¹₂B_ijdxⁱ ∧dx^j are the metric and the antisymmetric Kalb-Ramond 2-form on M, where (xⁱ) are local coordinates on it. Then g(X) =X^∗gandB(X) =X^∗Bare the pull-back metric andB-field by means of the mapX.

In what follows we ignore the dilaton coupling, which enters the action at linear order inα^′. Now we would like to find a gauged extension of the action (2.1) for as general X- dependent background fields g(X) and B(X) as possible. To this end we introduce a set of world sheet 1-forms A taking values in a Lie algebroid L →^ρ T M. Let us denote local sections ofL by ea, a= 1, . . . , r,r being the rank of the vector bundle L. These sections

4Cf. [13,21] for the analogous result in a less general context.

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are mapped via the smooth map ρ to a set of vector fields ρa: L →^ρ T M

ea 7→ ρ(ea) :=ρa=ρⁱ_a(X)∂i . (2.2) Recalling that Lis equipped with a bracket such that

[ea, eb]L=C_ab^c (X)ec, (2.3) and the mapρ is a homomorphism, we deduce that the vector fields ρ_a satisfy

[ρa, ρ_b] =C_ab^c (X)ρc, (2.4) the bracket being the standard Lie bracket of vector fields. As for the gauge fields A = A^ae_a∈Γ(L), these are mapped toρ(A)∈Γ(T M). In this subsection we examine the case when these gauge fields are introduced in the theory through minimal coupling. In other words we promote the world sheet differential to a covariant one as

DXⁱ = dXⁱ−ρⁱ_a(X)A^a, (2.5)

or in index-free notation DX = dX−ρ(A). The map ρ need not be invertible and in general it is not. For example, if we choose to gauge the theory only alongd≤n= dimM directions of the target, then in adapted coordinates where the set of spacetime indices{i}

splits into {µ, m}, for µ= 1, . . . , n−d and m = n−d+ 1, . . . , n, we would have ρ^µa = 0 and consequently DX^µ= dX^µ. In any case, minimal coupling then means that the gauge fields appear in the gauged theory only through the covariant derivative DXⁱ.

According to the above, the action we consider is Sm.c.[X, A] =

Z

Σ

1

2gij(X)DXⁱ∧ ∗DX^j+ Z

Σ

1

2Bij(X)DXⁱ∧DX^j . (2.6) Then the question is under which conditions does this action have a gauge symmetry generated by the vector fields ρa, namely

δǫXⁱ =ρⁱ_a(X)ǫ^a(σ), (2.7)

where ǫ^a ∈ C^∞(Σ) is the gauge transformation parameter. This crucially depends on how the gauge fields A^a transform. Let us consider a general Ansatz for their gauge transformation:

δǫA^a= dǫ^a+C_bc^a(X)A^bǫ^c+ ∆A^a, (2.8) where for the moment the additional term ∆A^a, which of course has to be a world sheet 1-form, is not specified. Now we can determine how the covariant derivative transforms under these gauge transformations. It is found that

δǫDXⁱ =ǫ^a∂jρⁱ_aDX^j−ρⁱ_a∆A^a . (2.9)

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Requiring a covariant transformation rule for the covariant derivative motivates us to refine the Ansatz by writing

∆A^a=ω_bi^a(X)ǫ^bDXⁱ+φ^a_bi(X)ǫ^b∗DXⁱ . (2.10) First of all, let us mention that the addition of the second term is possible only in two dimensions, where the Hodge dual of an 1-form is also an 1-form. Second, the coefficients ω_bi^a and φ^a_bi are still undetermined functions defined locally on M and their geometric interpretation will be clarified in due course.

Next let us examine the behaviour of the action (2.6) under the above gauge transfor- mations. A direct computation leads to the result

δǫS_m.c. = Z

Σ

ǫ^a 1

2(Lρag)_ijDXⁱ∧ ∗DX^j +1

2(LρaB)_ijDXⁱ∧DX^j

− Z

Σ

gijρⁱ_a∆A^a∧ ∗DX^j+Bijρⁱ_a∆A^a∧DX^j

. (2.11)

It is directly observed that in case ∆A^a = 0, namely when A^a transforms as a standard non-Abelian gauge field, the results reviewed in the introduction are immediately recovered.

However, now a new possibility is revealed. Due to (2.10) the terms in the second line of the transformed action can compensate for the ones in the first line. Indeed, defining the 1-forms ω_b^a =ω^a_bidxⁱ and φ^a_b =φ^a_bidxⁱ, the action functional is gauge invariant if and only if the following two conditions hold:

Lρag=ω_a^b∨ιρbg−φ^b_a∨ιρbB , (2.12) L_ρaB =ω_a^b∧ι_ρbB±φ^b_a∧ι_ρbg , (2.13) where the ± sign is due to∗² =∓1 and we recall that ∨ denotes the symmetrized tensor product; thus for example (ω^a_b∨ιρbg)ij =ω_ai^b ρ^k_bgkj+ω^b_ajρ^k_bgki. Clearly (2.12) is a symmetric equation and (2.13) an antisymmetric one.

We observe that the action (2.6) can be gauge invariant under the transformation (2.7) without the vector fields ρa being Killing, i.e. without them generating isometries. This is true provided the gauge fields transform according to (2.8). Thus the resulting invariance conditions are milder than in the case of conventional gauging. Furthermore, it is worth noting that due to the additional term in (2.8), which is proportional to the Hodge dual covariant differential, the invariance conditions mix the metric and the Kalb-Ramond field.

This is indicative of an interpretation in terms of generalized geometry, as we discuss immediately below.

2.2 Geometric interpretation and generalized Riemannian metric

Having found the extended invariance conditions (2.12) and (2.13) that guarantee gauge invariance of the action, we would like to explain the geometric role of the coefficients ω_bi^a and φ^a_bi. In order to understand their geometric interpretation let us examine what happens under a change of basis in the vector bundle L. First note that when we change the basisea→Λ(X)^b_aebthe gauge field transforms asA^a→(Λ⁻¹(X))^a_bA^b so thatA=A^aea

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remains invariant. Next, from the transformation of (2.3) we determine the transformation properties of the structure functions C_bc^a(X):

C_bc^a →(Λ⁻¹)^a_dΛ^e_bΛ^f_cC_ef^d + 2(Λ⁻¹)^a_dΛ^e_[bρⁱ_e∂iΛ^d_c], (2.14) where the underlined index does not participate in the antisymmetrization. We notice that the structure functions do not transform as tensors under the change of basis, as expected (cf. [14]).

Demanding that the form of the gauge transformation (2.8) does not change under the change of local basis we obtain the transformation properties of the coefficientsω_bi^a andφ^a_bi: ω^a_bi → (Λ⁻¹)^a_cω_di^c Λ^d_b −Λ^c_b∂_i(Λ⁻¹)^a_c, (2.15) φ^a_bi → (Λ⁻¹)^a_cφ^c_diΛ^d_b . (2.16) It is directly observed that φ^a_b transforms as a tensor, however ω^a_b does not; it transforms instead as a connection, as noticed before in [14]. This means that ω^a_bi are the coefficients of a connection 1-form onL:

∇^ωea=ω_a^b⊗eb, (2.17)

whilst φ^a_b are the components of an endomorphism-valued 1-formφ∈Γ(T^∗M ⊗E^∗⊗E).

Since the difference of any two connections on a vector bundle is such an endomorphism- valued 1-form on the manifoldM, this essentially means that we have two connections on L,∇=∇^ω and ∇e =∇^ω+φ. In fact it is more convenient to introduce the connections

∇^±=∇^ω±φ , (2.18)

for the reason that is explained immediately below.

In order to gain a better geometrical understanding of the mixing of g and B in the invariance conditions (2.12) and (2.13), it is useful to consider the two maps

E^± :=B±g:T M →T^∗M , (2.19)

defined via the interior product. Additionally, we define the combinations

(Ω^±)^a_b := (ω±φ)^a_b . (2.20)

Then the two invariance conditions for the Lorentzian signature of the world sheet metric may be expressed as

LρaE^± = (Ω^∓)^b_a⊗ιρbE^±−ιρbE^∓⊗(Ω^±)^b_a . (2.21) This expression highlights the role of the two connections defined above. Moreover, let us recall from [25,26] the role of the maps E^±. In generalized complex geometry one deals with structures on the generalized tangent bundle T M ⊕T^∗M, whose structure group is O(n, n). A generalized Riemannian metric is a reduction of the structure group to O(n)×O(n) or equivalently a choice of a n-dimensional subbundle C+ which is positive

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definite with respect to the natural inner product on T M ⊕T^∗M and whose negative- definite orthogonal complement is C−. Then (cf. Proposition 6.6 in [25]) the subbundles C_± are identified with the graphs of the maps E^±. Using these data one can define a generalized metric H : T M ⊕T^∗M → T M ⊕T^∗M on the generalized tangent bundle, which in terms ofg and B takes the form

H= −g⁻¹B g⁻¹ g−Bg⁻¹B Bg⁻¹

!

, (2.22)

where g⁻¹ is the inverse metric. We refer to [26] for more details, since we are not going to use this metric any more in this paper. However, it is important to keep in mind the appearance of the generalized tangent bundle since it will play an interesting role at the end of this section, after we discuss the inclusion of WZ terms in the theory.

2.3 Inclusion of WZ term and non-minimally coupled gauge fields

Let us now go one step further and include a WZ term in the σ-model action. Therefore the original ungauged action now is

S_0,WZ[X] = Z

Σ

1

2gij(X)dXⁱ∧ ∗dX^j+ Z

Σˆ

1

6Hijk(X)dXⁱ∧dX^j∧dX^k, (2.23) where ˆΣ is an open membrane world volume whose boundary is the world sheet Σ,∂Σ = Σ.ˆ As beforeH(X) =X^∗H, whereH = ¹₆Hijkdxⁱ∧dx^j∧dx^kis a closed 3-form onM. This 3- form need not be exact at the global level, thusH = dBonly locally. Before proceeding with our analysis, let us recall some well-known facts from the ordinary gauging of this action.

First of all, in the presence of WZ terms minimal coupling of the gauge fields is not sufficient to guarantee a gauge invariant action. This means that terms with “bare” gauge fields, i.e.

outside of covariant derivatives, have to be added to the topological sector of the action. On the other hand this is not necessary for the kinetic sector, where gauge fields can still enter through minimal coupling.⁵ Second, already in the isometric case the invariance conditions on the background fields do not automatically render the action gauge invariant. There are additional constraints that have to be met. This will be true also in our analysis here.

In order to address these issues in the more general context that we employ in this paper, we start from a candidate gauge action of the form

S_WZ[X, A] = Z

Σ

1

2gij(X)DXⁱ∧ ∗DX^j+ Z

Σˆ

H(X) + Z

Σ

A^a∧θa+ Z

Σ

1

2γabA^a∧A^b, (2.24) where θ_a= θ_ai(X)dXⁱ are 1-forms andγ_ab functions on the target space M, both pulled back byX: Σ→M. The covariant derivative DXⁱ is again defined as in (2.5), with gauge fields taking values in the vector bundleL. Once more we ask under which conditions this

5Of course it is possible to also consider non-minimally coupled kinetic terms but we do not see sufficient motivation for this in the context of the present paper. The investigation of the most general case will be presented in [2].

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action is invariant under the gauge transformations (2.7) and (2.8), which we repeat here for completeness:

δǫXⁱ = ρⁱ_a(X)ǫ^a(σ), (2.25)

δ_ǫA^a = dǫ^a+C_bc^a(X)A^bǫ^c+ω_bi^a(X)ǫ^bDXⁱ+φ^a_bi(X)ǫ^b∗DXⁱ . (2.26) Transforming the action (2.24) under these gauge transformations we find

δǫS_WZ = Z

Σ

ǫ^a 1

2

Lρag−ω_a^b∨ιρbg−φ^b_a∨θb

ijDXⁱ∧ ∗DX^j + +1

2

ιρaH−dθa+ω_a^b∧θb∓φ^b_a∧ιρbg

ijDXⁱ∧DX^j+

−

Lρaθb−C_ab^c θc+ιρb(ιρaH−dθa) + (γbd−ιρbθd)ω_a^d

idXⁱ∧A^b+ +

1

2Lρaγbc+C_ab^dγcd−1

2ιρcιρb(ιρaH−dθa) + (γbd−ιρbθd)ιρcω_a^d

A^b∧A^c

+ +

Z

Σ

(γ_ab−ι_ρaθ_b)(dǫ^a∧A^b+ǫ^cφ^b_ciDXⁱ∧ ∗A^a) . (2.27) Gauge invariance is established when all five lines in the above result vanish. However, the third and the fifth line together imply the fourth one, while the fifth one itself states that γ_ab =ιρaθ_b. Thus we conclude that the functional (2.24) is invariant with respect to gauge transformations of the form (2.25) and (2.26) if and only if the following four equations hold true:

Lρag=ω_a^b∨ιρbg+φ^b_a∨θb, (2.28) ι_ρaH = dθ_a−ω_a^b∧θ_b±φ^b_a∧ι_ρbg , (2.29) supplemented by the constraints

ιρaθb+ιρbθa = 0, (2.30)

Lρaθb = C_ab^dθd−ιρb(ιρaH−dθa) . (2.31) The last constraint can be rewritten in the form of a double contraction on H:

ιρbιρaH =C_ab^dθd+ dιρ_[aθ_b]−2Lρ_[aθ_b], (2.32) an expression which will acquire a natural geometric interpretation below.

When both ω_b^a and φ^a_b vanish, the known results for the non-Abelian but isometric case are recovered [8–11,21,22]. However we observe that in general the conditions (2.28) and (2.29) as well as the additional constraint (2.31) are milder than in the standard case.

(The constraint (2.30) remains the same though.)

Addendum on the geometric interpretation. The geometric interpretation of these results is similar to the minimally-coupled case; there are two connections ∇^± on L that control the gauging. However, in the present case the role of the generalized tangent bundle

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is further clarified. Indeed, in the spirit of (2.2) for the map ρ, the 1-form θa may be associated to the following mapθfrom the vector bundleLto the cotangent bundle of M:

L →^θ T^∗M

ea 7→ θ(ea) :=θa=θaidxⁱ . (2.33) Combining the two maps we obtain

L ^ρ⊕θ→ T M⊕T^∗M

ea 7→ (ρ⊕θ)(ea) :=ρa+θa=ρⁱ_a∂i+θaidxⁱ . (2.34) This is reminiscent of a Courant algebroid structure on the generalized tangent bundle, in particular the H-twisted standard Courant algebroid. Local sections of T M ⊕T^∗M are generalized vectors ξa=ρa+θa and the corresponding bracket of sections is the Courant bracket

[ξ_a, ξ_b] = [ρ_a, ρ_b] +L_ρaθ_b− L_ρbθ_a−1

2d (ι_ρaθ_b−ι_ρbθ_a)−ι_ρaι_ρbH . (2.35) Then the constraint (2.31) is interpreted as closure of thisH-twisted Courant bracket for the generalized vectors ξa. This is more easily verified using the equivalent expression (2.32).

Moreover, the constraint (2.30) is equivalent to the statement that the generalized vectors ξa have a vanishing non-degenerate symmetric bilinear form

hξa, ξ_bi=ιρaθ_b+ιρ_bθa . (2.36) In other words the two constraints require thatξaare sections of an involutive and isotropic subbundle ofT M⊕T^∗M. (This was also noticed in a similar context in [13] and also in [22], where in the presence of additional scalar fields it was possible to relax the isotropy con- dition.) Such subbundles are called (small) Dirac structures and will be studied in more detail in the upcoming section.

3 Connections and gauge symmetries for Dirac structures

3.1 Dirac structures and Dirac σ-models

In the previous section we found that the extended invariance conditions (2.28) and (2.29) on the background fields, which are necessary for gauge invariance, are controlled by the coefficients ω_bi^a and φ^a_bi which give rise to two connections on the vector bundleL. A nat- ural question is whether these coefficients can be determined explicitly, or in other words, whether we can really provide the corresponding connections. Furthermore, at the end of the last section we saw that the additional constraints obstructing the gauging of the σ- model are associated to a particular class of subbundles in theH-twisted Courant algebroid.

In order to study the construction of the two connections, we now focus on Dirac structures. As indicated already, these are maximal subbundles D⊂ T M ⊕T^∗M of the generalized tangent bundle with the following two properties:

[Γ(D),Γ(D)] ⊂Γ(D), (3.1)

hΓ(D),Γ(D)i = 0, (3.2)

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namely they are involutive with respect to the H-twisted Courant bracket and isotropic with respect to the symmetric bilinear form on the Courant algebroid. They were intro- duced in [27] as generalization of symplectic and Poisson structures. Maximality in this case means that their rank is half of the rank ofT M⊕T^∗M. Moreover, since maximality is not indicated by any of the found constraints, one may define a small Dirac structure as being any subbundle satisfying the above two conditions. This definition first appeared in [13].

At the availability of a metricgonM, which is the case here, yet another parametriza- tion of Dirac structures is possible, as explained in [23,24]. Instead of referring to the gen- eralized vectorsξ_a=ρ_a+θ_a, one uses the Riemannian metricgto identifyT M withT^∗M and introduces an orthogonal operatorO=Oⁱ_j∂i⊗dX^j ∈Γ(End(T M)) whose graph is the Dirac structureD. The role of this operator is that for any element, sayv⊕ηwithva vector and η an 1-form, of the Dirac structure there exists a unique sectiona of T M such that

v⊕η= (id− O)a⊕((id +O)a)^∗ , (3.3) where the notation υ^∗ and η^∗ for a vector field υ or a 1-form η denotes the action of the metric or inverse metric that results in an 1-form or a vector field respectively, explicitly

υ^∗= (υⁱ∂i)^∗ =υⁱgijdx^j, η^∗= (ηidxⁱ)^∗ =ηig^ij∂j . (3.4) Given a Dirac structure D one may uniquely construct a two-dimensional σ-model called Dirac σ-model (DSM) [23], which was introduced as a simultaneous generalization of the Poisson sigma model [28,29] and the G/G WZW model. Its action functional is

S_DSM[X, v⊕η] = Z

Σ

1

2g_ij(X)DXⁱ∧ ∗DX^j+ Z

Σ

η_i∧dXⁱ− 1 2η_i∧vⁱ

+

Z

Σˆ

H , (3.5) where v⊕η ∈ Ω¹(Σ, X^∗D) and here we defined the world sheet covariant differential as DXⁱ = dXⁱ−vⁱ. In terms of the alternative parametrization introduced above, i.e., here witha∈Ω¹(Σ, X^∗T M), an equivalent form of this action, already presented in ref. [23], is

S_DSM[X,a] = Z

Σ

1 2gij

dXⁱ−(id− O)ⁱ_ka^k

∧ ∗

dX^j−(id− O)^j_la^l

+ +

Z

Σ

(gij+Oij)a^j∧dXⁱ+Oijaⁱ∧a^j +

Z

Σˆ

H , (3.6)

with indices lowered by means of the metric g. Even though there is a kinetic sector, this model turns out to be a topological field theory [23]. A non-topological field theory can be constructed by assuming instead a small (non-maximal) Dirac structure [13].

An interesting observation, already suggested in [13], albeit in a less general context, is that upon the relations

v=ρ(A)⇒vⁱ=ρⁱ_a(X)A^a and η=θ(A)⇒ηi =θai(X)A^a, (3.7) the action functional (2.24) takes the form (3.5). This relation among the two action functionals is very useful in constructing the desired connections and determining the general form of the gauge symmetries for the DSM (3.5). The latter were suggested without proof in the original publication [23], but here we will prove them at the end of this section.

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3.2 Constructing the connections ∇^± on Dirac structures

Our interest now is to determine explicitly the two connections ∇^± in the case that both L=Dand its imageDe under the mapρ⊕θin (2.34) are Dirac structures in theH-twisted standard Courant algebroid.

First, it is useful to recall a lemma proven in [23] (Lemma 1), from which it follows that the operator

(id− O) +b(id +O), (3.8) whereb is positive or negative symmetric operator, is invertible. Then let us consider the sections v⊕η ∈ Ω¹(Σ, X^∗D) on the Dirac structure D and ev⊕ηe∈ Ω¹(Σ, X^∗D) on thee Dirac structure D. Recalling the parametrization in terms of the orthogonal operator, wee parametrizeD byO and De by O. These considerations allow us to writee

v= (id− O)a, η = ((id +O)a)^∗ , (3.9) and

e

v = (id−O)ae , ηe=

(id +O)ae ∗

, (3.10)

for somea∈Ω¹(Σ, X^∗T M). Moreover by assumption it holds that

ev=ρ(v⊕η), and ηe=θ(v⊕η) . (3.11) These relations directly translate into

id−Oe =ρ((id− O) + (id +O)^∗), (3.12) id +Oe = (θ((id− O) + (id +O)^∗))^∗ , (3.13) which in turn yield

1

2(θ^∗+ρ) ((id− O) + (id +O)^∗) = id, (3.14) 1

2(θ^∗−ρ) ((id− O) + (id +O)^∗) = Oe . (3.15) Now the right hand side of both equations is an invertible operator. Thus we conclude that the operators

θ^∗±ρ:D→T M , (3.16)

are invertible too, with inverses denoted as (θ^∗±ρ)⁻¹:T M →D. We note in passing that certainly none of the maps ρand θ^∗ are required to be invertible separately.

Now let us state and prove the main result of this section. For the vector bundles D and De as above, the invariance conditions (2.28) and (2.29) are solved by the coefficients

ω_bi^a = Γ^a_bi−φ^a_bi+T_bi^a, (3.17) φ^a_bi = [(θ^∗−ρ)⁻¹]^a_k

∇˚iρ^k_b −ρ^k_cT_bi^c

, (3.18)

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where Γ^a_biare the coefficients of the Levi-Civita connection∇^LC onD, ˚∇is the Levi-Civita connection on T M and

T_bi^a = [(θ^∗+ρ)⁻¹]^a_k

∇˚i(θ^∗+ρ)^k_b − 1 2ρ^l_bH_li^k

, (3.19)

where one index of the 3-form H is raised by means of the metric g. This is proven by direct computations. Indeed the first invariance condition (2.28) holds because

(Lρag)_ij = ρ^k_a∂kgij + 2g_k(i∂_j)ρ^k_a

= ρ^k_a∇˚kgij + 2ρ^k_aΓ^l_k(ig_j)l+ 2g_k(i˚∇_j)ρ^k_a−2g_k(iΓ^k_j)lρ^l_a+ 2g_k(iΓ^b_j)aρ^k_b

= 2g_k(i

∇˚_j)ρ^k_a+ Γ^b_j)aρ^k_b

= 2g_k(i

(θ^∗−ρ)^k_bφ^b_aj)+ρ^k_bT_aj)^b +ρ^k_bω^b_aj)+ρ^k_bφ^b_aj)−ρ^k_bT_aj)^b

= 2g_k(i

(θ^∗)^k_bφ^b_aj)+ρ^k_bω_aj)^b

=

ω^b_a∨ιρbg+φ^b_a∨θb

ij , (3.20)

as required. The second condition is proven as shown below, for Lorentzian world sheets:

ιρaH−dθa+ω_a^b∧θ_b+φ^b_a∧ιρ_bg

ij =

(3.17)

= ρ^k_aH_kij −2∂_[iθ_aj]+ 2(Γ^b_a[i+T_a[i^b )θ_bj]−2φ^b_a[i(θ−ιρg)_bj]=

(3.18)

= ρ^k_aH_kij −2˚∇_[i(θ^∗+ρ)^k_ag_kj]+ 2T_a[i^b θ_bj]+ 2ρ^k_cT_a[i^c g_kj]=

(3.19)

= 0, (3.21)

as required.

Having found the coefficients ω^a_bi and φ^a_bi, it is now simple to write down the two connections∇^±. LetT(ρ) =T_a^b⊗e^a⊗ρb ∈Γ(T^∗M⊗D^∗⊗T M) and denote by the same letters (θ^∗±ρ)⁻¹ ∈Γ(T^∗M ⊗D) the sections of the indicated bundle that correspond to the operators (θ^∗±ρ)⁻¹. Then

∇^ω = ∇^LC−φ+T , (3.22)

φ = ι_{(˚∇−T)(ρ)}(θ^∗−ρ)⁻¹, (3.23) where the contraction is among the single T M and T^∗M indices of the corresponding sections. This leads directly to

∇⁺=∇^LC+T , (3.24)

∇⁻=∇^LC+T −2ι_{(˚∇−T)(ρ)}(θ^∗−ρ)⁻¹ . (3.25) We close this section by providing an alternative formulation of the main result. In particular, having parametrized the Dirac structures in terms of orthogonal operators with the aid of the metricg, it is useful to express the connections∇^± in terms of the operator

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O whose graph is D. Clearly this is more transparent in the parametrization in terms of the unconstrained field a = aⁱ∂i ∈ Ω¹(Σ, X^∗T M). Its gauge transformation δǫaⁱ = dǫⁱ+C_jkⁱ a^jǫ^k+ω_jkⁱ ǫ^jDX^k+φⁱ_jkǫ^j∗DX^k involves then the coefficients

ωⁱ_jk = Γⁱ_jk+1

2(O⁻¹)_mⁱ∇jO^m_k+1

8(id +O⁻¹)_mⁱH_jl^m(id− O)^l_k, (3.26) φⁱ_jk = −1

2(O⁻¹)_mⁱ∇jO^m_k+1

8(id− O⁻¹)_mⁱH_jl^m(id− O)^l_k . (3.27) In coordinate independent form the connections ∇^± are

∇⁺ = ∇^LC+1

2H(id− O) , (3.28)

∇⁻ = ∇^LC−ι_(˚∇−1

2H)(id−O)O⁻¹, (3.29)

where H(id− O) = ¹₂H_jkⁱ dx^j ⊗(id− O)^l_i∂_l⊗dx^k ∈ Γ(T^∗M ⊗T M⊗T^∗M) and O⁻¹ ∈ Γ(T^∗M⊗T M). The contracted indices are the T^∗M index of O with the T M index of (˚∇ − ¹₂H)(id− O). Then along withδǫX = (id− O)ǫ, the action functional (3.5) is gauge invariant. As a final note, let us observe that in case φ = 0, namely when there is no term proportional to ∗DXⁱ in the gauge transformation of the gauge field, there is only one connection on Dand the result depends solely on θ^∗+ρ.

3.3 General form of the gauge symmetries for DSMs

Now we would like to finally determine the gauge symmetries for the DSM and show that they take the form suggested without proof in [23]. Note that we specialize here to the case ∗²= 1 as in the aforementioned paper.

Here we work in the parametrization of the Dirac structure in terms of the operator O, using the unconstrained gauge field a ∈Ω¹(Σ, X^∗T M) instead of the constrained ones v⊕η∈Ω¹(Σ, X^∗D). Our purpose is then to determineδǫa. First of all, note that

δǫa= δǫaⁱ+ Γⁱ_jka^jδǫX^k

∂i= δǫaⁱ+ Γⁱ_jka^j(id− O)^k_lǫ^l

∂i, (3.30) sinceδǫX = (id− O)ǫ. Let us also recall that the gauge transformation for aⁱ is given as

δ_ǫaⁱ = dǫⁱ+C_jkⁱ (X)a^jǫ^k+ ∆aⁱ, (3.31) where ∆aⁱ =ωⁱ_jk(X)ǫ^jDX^k+φⁱ_jk(X)ǫ^j ∗DX^k. Two additional relations that will prove useful are the form of the pull-back ˚∇^∗ of the Levi-Civita connection ˚∇toX^∗T M,

˚∇^∗ǫ=

dǫⁱ+ Γⁱ_jkdX^jǫ^k

∂i, (3.32)

and the expression of the structure functions C_jkⁱ in terms of the operator O, which is derived in the appendix:

C_jkⁱ = 2(id− O)^l_[jΓⁱ_k]l+g^ilgmnOⁿ_[j˚∇lO^m_k]+1

2H_mnⁱ (id− O)^m_j(id− O)ⁿ_k, (3.33)

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where the weighted antisymmetrization is applied only on the indicesjandk. Substituting the above in (3.30) we obtain

δǫa = ˚∇^∗ǫ+

−Γⁱ_jkdX^jǫ^k+ (id− O)^l_jΓⁱ_kla^jǫ^k+ ∆aⁱ

∂i+ +

g^ilg_mn(Oⁿ_[j∇˚_lO^m_k]+1

2g^ilH_lmn(id− O)^m_j(id− O)ⁿ_k

a^jǫ^k∂_i . (3.34) Using that DXⁱ= dXⁱ−(id− O)ⁱ_ja^j and rearranging terms, we obtain

δǫa= ˚∇^∗ǫ−g

O⁻¹˚∇(O)a, ǫ∗

+ (∆aⁱ−Γⁱ_jkDX^jǫ^k)∂i+1

2H((id− O)a,(id− O)ǫ,·)^∗ . (3.35) The last step amounts to substituting the expressions for ω_jkⁱ and φⁱ_jk given in (3.26) and (3.27) in ∆aⁱ. A little algebra leads to the final result for the general form of the gauge transformation ofa:

δǫa= ˚∇^∗ǫ−g O⁻¹∇(O)a, ǫ˚ ∗

+ 1

2H((id− O)a,(id− O)ǫ,·)^∗+ + Θ

1

2H(DX,(id− O)ǫ,·)^∗+ (1− ∗)˚∇_DX(O)ǫ

, (3.36)

where we defined the operator Θ = 1

4(id +O⁻¹) +1

4(id− O⁻¹)∗ (3.37)

onT^∗Σ⊗X^∗T M. This operator is the inverse of the operator (id +O) + (id− O)∗, denoted byT in ref. [23]. This is also proven in the appendix. Then (3.36) is indeed the desired form for the gauge transformation. Note that additional trivial gauge symmetries parametrized by an endomorphism M ∈Γ(End(T^∗Σ⊗X^∗T M)), as explained in [23], may be included without any problem.

4 Application: Wess-Zumino Poisson σ-model

The Poisson sigma model is a typical example in the class of topological DSMs, whose target space is a Poisson manifold. An instructive way to obtain the model has ordinary 2D Yang-Mills theory as a starting point [29]:

S_YM[A] = Z

Σ

1

2F ∧ ∗F , (4.1)

where F = (dA_i +¹₂C_i^jkA_j ∧A_k)eⁱ is the field strength of the gauge field A = A_ieⁱ and {eⁱ} are the generators of the gauge algebra. In the first order formalism we introduce conjugate variablesXⁱ to the gauge fieldAi and rewrite the action as

S_YM[X, A] = Z

Σ

1 2

XⁱFi−1

2(Xⁱ)²∗1

. (4.2)